1. Field of the Invention
The invention relates to computer modeling of physical objects by the manual inputting of shapes and the display of physical quantities. More particularly, the invention relates to generation of finite elements for representing the shape of the objects.
2. Background Art
Finite element modeling (FEM) is one of the most widely used computer aided engineering tools applied to solve engineering problems governed by partial differential equations. Such problems include heat transfer, stress and vibration analysis, diffusion, fluid flow, and electrical and magnetic fields. Finite element modeling is particularly important when the shapes of the objects to be modeled are relatively complex since the differential equations become increasingly difficult to solve or even to approximate.
In solving such an engineering problem by finite element modeling, there is an iterative interaction between the designer and the computer. First the shape of the physical object is defined, usually by the manual inputting of the outlines of the shape. Then, this shape is broken down into finite elements. Each finite element is small enough that the desired physical quantities can be approximated by suitable interpolating functions over a single finite element. Finally, one or more difference equations, which are a finite element form of the relevant set of partial differential equations, are solved by the computer based upon the generated finite elements. The results of the difference equations for one or more physical quantities are displayed. The user inspects these displayed quantities, adjusts the shape, and repeats the finite element generation and the difference equation solution in hopes of obtaining better results according to some criterion known to the user.
An important and time consuming part of finite element modeling is the decomposition of the problem domain (object shape) into the finite elements. It is important to automate this decomposition process for several reasons. Manual finite element generation is a tedious process prone to error if the shape is complex. Finite element generation is often the rate limiting step in which the designer proposes a design, analyzes it, and, based on the analysis, modifies it. As much as 80% of the analyst's time may be occupied with the generation of the finite elements. Therefore, automating the finite element generation improves the design cycle time. Also, a reliable, automatic finite element generator is a prerequisite for an automated design optimization system.
Finite elements are usually generated using a two step process. First, in a coarse decomposition, the object is partitioned into disjoint subdomains. Then, in a fine decomposition, each subdomain is further partitioned into finite elements. This invention is concerned with fine decomposition and is intended for use in conjunction with the general type of coarse decomposition disclosed by Nackman and Srinivasan, two of the present inventors, in U.S. patent application, Ser. No. 717,368, filed Mar. 25, 1985, now abandoned, and its continuation-in-part, Ser. No. 97,382, filed Sept. 16, 1987 and now U.S. Pat. No. 4,797,842 incorporated herein by reference.
The goal of finite element modeling is to obtain an approximate solution to a system of partial differential equations (with boundary and initial conditions) over some domain. The approach is to decompose the domain into subdomains, called elements, and then, using the idea of interpolation, to seek an approximate solution for the dependent variables. To ensure accuracy, the elements in regions in which the dependent variables change rapidly should be small and many. Such rapid changes occur: (1) in regions of rapid changes in geometrical shapes, e.g., near reentrant corners; (2) in regions of rapid changes in the boundary conditions, e.g., near a concentrated boundary heat source; and (3) in regions of rapid changes in material properties, e.g., at the interface between two bonded materials. On the other hand, for computational efficiency and, to a lesser extent, accuracy, the elements in regions in which the dependent variable changes slowly should be large and few.
The purpose of an automatic finite element generator is to decompose automatically a model of the shape of a physical object into a collection of elements that provides an appropriate balance between accuracy and efficiency. The set of boundaries between the generated finite elements is often referred to as a mesh. Mesh generation techniques have been surveyed by M. S. Shephard in a technical article entitled "Finite Element Modeling Within an Integrated Geometric Modeling Environment: Part I--Mesh Generation" appearing in Engineering with Computers, vol. 1, 1985 at pages 61-71.
Finite element generation schemes are classified here according to what user interaction is required. In a manual scheme, both coarse and fine decomposition require user interaction. In a semi-automatic scheme, the coarse decomposition requires user interaction, but the fine decomposition does not. In an automatic scheme, neither the fine nor the coarse decomposition requires user interaction other than the specification of a few parameters.